Higher Dimensions
More general Heisenberg groups Hn may be defined for higher dimensions in Euclidean space, and more generally on symplectic vector spaces. The simplest general case is the real Heisenberg group of dimension 2n+1, for any integer n ≥ 1. As a group of matrices, Hn (or Hn(R) to indicate this is the Heisenberg group over the ring R or real numbers) is defined as the group of square matrices of size n+2 with entries in R:
where
- a is a row vector of length n,
- b is a column vector of length n,
- is the identity matrix of size n.
Read more about this topic: Heisenberg Group
Famous quotes containing the words higher and/or dimensions:
“The lesson learned here is a costly one: If you stand up for your principles, follow the law, and win massively, you lose totally.”
—Linda J. Carpenter, U.S. educator. As quoted in the Chronicle of Higher Education, p. A38 (July 15, 1992)
“It seems to me that we do not know nearly enough about ourselves; that we do not often enough wonder if our lives, or some events and times in our lives, may not be analogues or metaphors or echoes of evolvements and happenings going on in other people?—or animals?—even forests or oceans or rocks?—in this world of ours or, even, in worlds or dimensions elsewhere.”
—Doris Lessing (b. 1919)